Tag Archives: geometry

Geometry and art, an exhibition in Toronto (Canada)

I received this notice from ArtSci Salon mailing (on February 7, 2020 via email),

Geometry is Life

Robin Kingsburgh

February 5 — 16, 2020
Opening Reception: Saturday, February 8, 2 — 5 pm​

Cicada (detail), Robin Kingsburgh (Acrylic on MDF board, 36″ x 38″, 2018)

My work takes inspiration from geometry. For me the square and the circle are starting points. And ending points. The square, defined by the horizontal and the vertical: it’s all you need. The circle: a snake biting its tail; the beginning and end; the still point. Geometric archetypes. But there is no perfect circle; there is no perfect square. The beauty of Pythagoras is within our minds. Rendered by the human hand, the square becomes imperfect, and becomes a part of the human world – where imperfection reigns. The rhythm of imperfection is beauty, where order and chaos dance, and sometimes balance.

Robin Kingsburgh is a trained astronomer (Ph.D. in Astronomy, 1992, University College London). Her artistic education comes from studies at University of Toronto, as well as in the U.K. and France, and has paralleled her scientific development. She currently teaches various Natural Science courses at York University, Toronto. Her scientific background influences her artwork in an indirect, subconscious way, where she employs geometric motifs as a frequent theme. She is a member of Propeller Gallery, where she shows her artwork on a regular basis. She has recently been elected to the Ontario Society of Artists.

There you have it. Have a nice weekend!

ETA February 10, 2020: I’m sorry I forgot to include the address: Propeller Gallery, 30 Abell St Toronto. Wed-Sat 12-6pm, Sun 12-5pm

Fusing graphene flakes for 3D graphene structures that are 10x as strong as steel

A Jan. 6, 2017 news item on Nanowerk describes how geometry may have as much or more to do with the strength of 3D graphene structures than the graphene used to create them,

A team of researchers at MIT [Massachusetts Institute of Technology] has designed one of the strongest lightweight materials known, by compressing and fusing flakes of graphene, a two-dimensional form of carbon. The new material, a sponge-like configuration with a density of just 5 percent, can have a strength 10 times that of steel.

In its two-dimensional form, graphene is thought to be the strongest of all known materials. But researchers until now have had a hard time translating that two-dimensional strength into useful three-dimensional materials.

The new findings show that the crucial aspect of the new 3-D forms has more to do with their unusual geometrical configuration than with the material itself, which suggests that similar strong, lightweight materials could be made from a variety of materials by creating similar geometric features.

The findings are being reported today [Jan. 6, 2017\ in the journal Science Advances, in a paper by Markus Buehler, the head of MIT’s Department of Civil and Environmental Engineering (CEE) and the McAfee Professor of Engineering; Zhao Qin, a CEE research scientist; Gang Seob Jung, a graduate student; and Min Jeong Kang MEng ’16, a recent graduate.

A Jan. 6, 2017 MIT news release (also on EurekAlert), which originated the news item, describes the research in more detail,

Other groups had suggested the possibility of such lightweight structures, but lab experiments so far had failed to match predictions, with some results exhibiting several orders of magnitude less strength than expected. The MIT team decided to solve the mystery by analyzing the material’s behavior down to the level of individual atoms within the structure. They were able to produce a mathematical framework that very closely matches experimental observations.

Two-dimensional materials — basically flat sheets that are just one atom in thickness but can be indefinitely large in the other dimensions — have exceptional strength as well as unique electrical properties. But because of their extraordinary thinness, “they are not very useful for making 3-D materials that could be used in vehicles, buildings, or devices,” Buehler says. “What we’ve done is to realize the wish of translating these 2-D materials into three-dimensional structures.”

The team was able to compress small flakes of graphene using a combination of heat and pressure. This process produced a strong, stable structure whose form resembles that of some corals and microscopic creatures called diatoms. These shapes, which have an enormous surface area in proportion to their volume, proved to be remarkably strong. “Once we created these 3-D structures, we wanted to see what’s the limit — what’s the strongest possible material we can produce,” says Qin. To do that, they created a variety of 3-D models and then subjected them to various tests. In computational simulations, which mimic the loading conditions in the tensile and compression tests performed in a tensile loading machine, “one of our samples has 5 percent the density of steel, but 10 times the strength,” Qin says.

Buehler says that what happens to their 3-D graphene material, which is composed of curved surfaces under deformation, resembles what would happen with sheets of paper. Paper has little strength along its length and width, and can be easily crumpled up. But when made into certain shapes, for example rolled into a tube, suddenly the strength along the length of the tube is much greater and can support substantial weight. Similarly, the geometric arrangement of the graphene flakes after treatment naturally forms a very strong configuration.

The new configurations have been made in the lab using a high-resolution, multimaterial 3-D printer. They were mechanically tested for their tensile and compressive properties, and their mechanical response under loading was simulated using the team’s theoretical models. The results from the experiments and simulations matched accurately.

The new, more accurate results, based on atomistic computational modeling by the MIT team, ruled out a possibility proposed previously by other teams: that it might be possible to make 3-D graphene structures so lightweight that they would actually be lighter than air, and could be used as a durable replacement for helium in balloons. The current work shows, however, that at such low densities, the material would not have sufficient strength and would collapse from the surrounding air pressure.

But many other possible applications of the material could eventually be feasible, the researchers say, for uses that require a combination of extreme strength and light weight. “You could either use the real graphene material or use the geometry we discovered with other materials, like polymers or metals,” Buehler says, to gain similar advantages of strength combined with advantages in cost, processing methods, or other material properties (such as transparency or electrical conductivity).

“You can replace the material itself with anything,” Buehler says. “The geometry is the dominant factor. It’s something that has the potential to transfer to many things.”

The unusual geometric shapes that graphene naturally forms under heat and pressure look something like a Nerf ball — round, but full of holes. These shapes, known as gyroids, are so complex that “actually making them using conventional manufacturing methods is probably impossible,” Buehler says. The team used 3-D-printed models of the structure, enlarged to thousands of times their natural size, for testing purposes.

For actual synthesis, the researchers say, one possibility is to use the polymer or metal particles as templates, coat them with graphene by chemical vapor deposit before heat and pressure treatments, and then chemically or physically remove the polymer or metal phases to leave 3-D graphene in the gyroid form. For this, the computational model given in the current study provides a guideline to evaluate the mechanical quality of the synthesis output.

The same geometry could even be applied to large-scale structural materials, they suggest. For example, concrete for a structure such a bridge might be made with this porous geometry, providing comparable strength with a fraction of the weight. This approach would have the additional benefit of providing good insulation because of the large amount of enclosed airspace within it.

Because the shape is riddled with very tiny pore spaces, the material might also find application in some filtration systems, for either water or chemical processing. The mathematical descriptions derived by this group could facilitate the development of a variety of applications, the researchers say.

“This is an inspiring study on the mechanics of 3-D graphene assembly,” says Huajian Gao, a professor of engineering at Brown University, who was not involved in this work. “The combination of computational modeling with 3-D-printing-based experiments used in this paper is a powerful new approach in engineering research. It is impressive to see the scaling laws initially derived from nanoscale simulations resurface in macroscale experiments under the help of 3-D printing,” he says.

This work, Gao says, “shows a promising direction of bringing the strength of 2-D materials and the power of material architecture design together.”

There’s a video describing the work,

Here’s a link to and a citation for the paper,

The mechanics and design of a lightweight three-dimensional graphene assembly by Zhao Qin, Gang Seob Jung, Min Jeong Kang, and Markus J. Buehler. Science Advances  06 Jan 2017: Vol. 3, no. 1, e1601536 DOI: 10.1126/sciadv.1601536  04 January 2017

This paper appears to be open access.

Handling massive digital datasets the quantum way

A Jan. 25, 2016 news item on phys.org describes a new approach to analyzing and managing huge datasets,

From gene mapping to space exploration, humanity continues to generate ever-larger sets of data—far more information than people can actually process, manage, or understand.

Machine learning systems can help researchers deal with this ever-growing flood of information. Some of the most powerful of these analytical tools are based on a strange branch of geometry called topology, which deals with properties that stay the same even when something is bent and stretched every which way.

Such topological systems are especially useful for analyzing the connections in complex networks, such as the internal wiring of the brain, the U.S. power grid, or the global interconnections of the Internet. But even with the most powerful modern supercomputers, such problems remain daunting and impractical to solve. Now, a new approach that would use quantum computers to streamline these problems has been developed by researchers at [Massachusetts Institute of Technology] MIT, the University of Waterloo, and the University of Southern California [USC}.

A Jan. 25, 2016 MIT news release (*also on EurekAlert*), which originated the news item, describes the theory in more detail,

… Seth Lloyd, the paper’s lead author and the Nam P. Suh Professor of Mechanical Engineering, explains that algebraic topology is key to the new method. This approach, he says, helps to reduce the impact of the inevitable distortions that arise every time someone collects data about the real world.

In a topological description, basic features of the data (How many holes does it have? How are the different parts connected?) are considered the same no matter how much they are stretched, compressed, or distorted. Lloyd [ explains that it is often these fundamental topological attributes “that are important in trying to reconstruct the underlying patterns in the real world that the data are supposed to represent.”

It doesn’t matter what kind of dataset is being analyzed, he says. The topological approach to looking for connections and holes “works whether it’s an actual physical hole, or the data represents a logical argument and there’s a hole in the argument. This will find both kinds of holes.”

Using conventional computers, that approach is too demanding for all but the simplest situations. Topological analysis “represents a crucial way of getting at the significant features of the data, but it’s computationally very expensive,” Lloyd says. “This is where quantum mechanics kicks in.” The new quantum-based approach, he says, could exponentially speed up such calculations.

Lloyd offers an example to illustrate that potential speedup: If you have a dataset with 300 points, a conventional approach to analyzing all the topological features in that system would require “a computer the size of the universe,” he says. That is, it would take 2300 (two to the 300th power) processing units — approximately the number of all the particles in the universe. In other words, the problem is simply not solvable in that way.

“That’s where our algorithm kicks in,” he says. Solving the same problem with the new system, using a quantum computer, would require just 300 quantum bits — and a device this size may be achieved in the next few years, according to Lloyd.

“Our algorithm shows that you don’t need a big quantum computer to kick some serious topological butt,” he says.

There are many important kinds of huge datasets where the quantum-topological approach could be useful, Lloyd says, for example understanding interconnections in the brain. “By applying topological analysis to datasets gleaned by electroencephalography or functional MRI, you can reveal the complex connectivity and topology of the sequences of firing neurons that underlie our thought processes,” he says.

The same approach could be used for analyzing many other kinds of information. “You could apply it to the world’s economy, or to social networks, or almost any system that involves long-range transport of goods or information,” says Lloyd, who holds a joint appointment as a professor of physics. But the limits of classical computation have prevented such approaches from being applied before.

While this work is theoretical, “experimentalists have already contacted us about trying prototypes,” he says. “You could find the topology of simple structures on a very simple quantum computer. People are trying proof-of-concept experiments.”

Ignacio Cirac, a professor at the Max Planck Institute of Quantum Optics in Munich, Germany, who was not involved in this research, calls it “a very original idea, and I think that it has a great potential.” He adds “I guess that it has to be further developed and adapted to particular problems. In any case, I think that this is top-quality research.”

Here’s a link to and a citation for the paper,

Quantum algorithms for topological and geometric analysis of data by Seth Lloyd, Silvano Garnerone, & Paolo Zanardi. Nature Communications 7, Article number: 10138 doi:10.1038/ncomms10138 Published 25 January 2016

This paper is open access.

ETA Jan. 25, 2016 1245 hours PST,

Shown here are the connections between different regions of the brain in a control subject (left) and a subject under the influence of the psychedelic compound psilocybin (right). This demonstrates a dramatic increase in connectivity, which explains some of the drug’s effects (such as “hearing” colors or “seeing” smells). Such an analysis, involving billions of brain cells, would be too complex for conventional techniques, but could be handled easily by the new quantum approach, the researchers say. Courtesy of the researchers

Shown here are the connections between different regions of the brain in a control subject (left) and a subject under the influence of the psychedelic compound psilocybin (right). This demonstrates a dramatic increase in connectivity, which explains some of the drug’s effects (such as “hearing” colors or “seeing” smells). Such an analysis, involving billions of brain cells, would be too complex for conventional techniques, but could be handled easily by the new quantum approach, the researchers say. Courtesy of the researchers

*’also on EurekAlert’ text and link added Jan. 26, 2016.

The geometry of baking with Alex Bellos and Evil Mad Scientist Laboratories

Alex Bellos has written some of my* favourite posts at the UK’s Guardian science blogs (for example, my Dec. 18,2012 posting about Bellos’ discussion of the math genius Ramanujan and my Oct. 17, 2012 posting about Bellos’ exploration of mathematics as a spiritual practice in Japan). Earlier this week in a June 26, 2013 posting Bellos wrote about a ‘new craze’, edible mathematics,  in a way that makes me wish I could revisit grade 10 geometry but with a good teacher this time (Note: Links have been removed),

When you slice a cone the surface produced is either a circle, an ellipse, a parabola or a hyperbola.

These curves are known as the conic sections.

And when you slice a scone in the shape of a cone, you get a sconic section – the latest craze in edible mathematics, a vibrant new culinary field.

On their fabulous website, the folk at Evil Mad Scientist provide a step-by-step guide to baking the sconic sections.

Here’s a sconic section image from the June 25, 2013 ‘Sconic sections’ posting by Lenore on the Evil Mad Scientist website,

To highlight the shapes even further, you can color in the cut surfaces with your favorite scone topping. Here, raspberry preserves show off a hyperbolic cut. [downloaded from http://www.evilmadscientist.com/2013/sconic-sections/]

To highlight the shapes even further, you can color in the cut surfaces with your favorite scone topping. Here, raspberry preserves show off a hyperbolic cut. [downloaded from http://www.evilmadscientist.com/2013/sconic-sections/]

Bellos highlights other edible mathematics projects including Maths on Toast and Dashing Bean but since I’ve always loved Escher I’m going to feature one of the other projects Bellos mentions, George Hart and his Möbius bagel,

After being cut, the two halves can be moved but are still linked together, each passing through the hole of the other.   (So when you buy your bagels, pick ones with the biggest holes.) [downloaded from http://georgehart.com/bagel/bagel.html]

After being cut, the two halves can be moved but are still linked together, each passing through
the hole of the other. (So when you buy your bagels, pick ones with the biggest holes.) [downloaded from http://georgehart.com/bagel/bagel.html]

Hart is serious about his Möbius bagel, from the bagel posting (Note: A link has been removed),

It is much more fun to put cream cheese on these bagels than on an ordinary bagel. In additional to
the intellectual stimulation, you get more cream cheese, because there is slightly more surface area.

Topology problem: Modify the cut so the cutting surface is a one-twist Mobius strip.
(You can still get cream cheese into the cut, but it doesn’t separate into two parts.)

Calculus problem: What is the ratio of the surface area of this linked cut
to the surface area of the usual planar bagel slice?

Note: I have had my students do this activity in my Computers and Sculpture class.  It is very successful if the students work in pairs, with two bagels per team.  For the first bagel, I have them draw the indicated lines with a “sharpie”.  Then they can do the second bagel without the lines. (We omit the schmear of cream cheese.) After doing this, one can better appreciate the stone carving of Keizo Ushio, who makes analogous cuts in granite to produce monumental sculpture.

Hope you enjoyed this mathematical ‘amuse-bouche’. If you want more, Bellos has included a few ‘how to’ videos, as well as, other images and links to websites in his posting.

* ‘my’ added to sentence on Aug. 12, 2015.